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Calculating Confidence Intervals for the Mean of a Lognormally Distributed Variable

USDA-ARS, Environmental Chemistry Lab., Bldg. 008, Room 102, BARC-West, Beltsville, MD 20705

* Corresponding author.

ABSTRACT

Many physical, chemical, and biological properties of soils exhibit skewed distributions that can be approximated by the two-parameter lognormal distribution. Recent attention in the soils literature has focused on the best method of estimating the mean for lognormally distributed variables; however, little attention has been given to methods for constructing confidence intervals about the mean of such variables. This paper reports on the efficacy of five different methods, one exact and four approximate, for constructing confidence intervals about the mean of a lognormally distributed variable. These methods were examined over a range of sample sizes (n = 6–100) by using Monte Carlo simulation to draw samples from four lognormal distributions with known mean and variances (coefficients of variation of 50, 100, 200, and 500%). Performance of the confidence-interval methods were assessed by identifying how close the calculated probability levels for the upper and lower confidence limits came to the stated probability levels. Two commonly applied methods, the asymptotic and Patterson's methods, failed to provide accurate probabilistic coverage. A nonparametric method developed by the authors was superior to both of these methods and compared favorably with Land's method in some cases. The latter is an exact method that gives the correct probabilistic coverage. The third method, attributed to Cox, was generally inferior to Land's and our nonparametric method, although it outperformed the asymptotic and Patterson's methods. For precise work, Land's method should be employed for constructing confidence intervals about the mean of a lognormally distributed variable.

Contribution from the USDA-ARS, Environmental Chemistry Lab. and The Upjohn Co., Kalamazoo, MI.

Received for publication January 27, 1989.

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